A395775 The twisted Euler phi-function for the Dirichlet character kronecker(-20,.) mod 20.

1, 2, 2, 4, 5, 4, 6, 8, 6, 10, 12, 8, 14, 12, 10, 16, 18, 12, 20, 20, 12, 24, 22, 16, 25, 28, 18, 24, 28, 20, 32, 32, 24, 36, 30, 24, 38, 40, 28, 40, 40, 24, 42, 48, 30, 44, 46, 32, 42, 50, 36, 56, 54, 36, 60, 48, 40, 56, 60, 40, 60, 64, 36, 64, 70, 48, 66, 72, 44, 60

Offset

1,2

Comments

We have h(-20*n^2), the class number of the order of Z[sqrt(-5)] with discriminant -20*n^2, is equal to 2*a(n).

In general, let d < 0 be a fundamental discriminant and n be a positive integer such that d*n^2 < -4, then h(d*n^2) = L(0,kronecker(d,.)) * n * Product_{p|n, p prime} (1 - kronecker(d,p)/p). This is a combination of Theorem 7.24 in "Primes of the form x^2+ny^2" by Cox and the Dirichlet class number formula. Here L(0,kronecker(-20,.)) = 2.

References

D. A. Cox, Primes of the form x^2+ny^2, Wiley, New York, 1989, p. 146.

Links

Jianing Song, Table of n, a(n) for n = 1..10000

Formula

Multiplicative with a(2^e) = 2^e, a(5^e) = 5^e, a(p^e) = p^(e-1)*(p-1) if p == 1, 3, 7, 9 (mod 20) and a(p^e) = p^(e-1)*(p+1) if p == 11, 13, 17, 19 (mod 20).

Example

Given L(0,kronecker(-20,.)) = 2, we have: h(-20*1^2) = 2*a(1) = 2, h(-20*2^2) = 2*a(2) = 4, h(-20*3^2) = 2*a(3) = 4, h(-20*4^2) = 2*a(4) = 8, ...

Mathematica

f[p_, e_] := p^(e-1)*(p - KroneckerSymbol[-20, p]); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, May 07 2026 *)

Prog

(PARI) a(n, {d=-20}) = my(f=factor(n)[, 1]~, w=#f); n * prod(i=1, w, 1 - kronecker(d, f[i])/f[i])

Crossrefs

A289741 is the corresponding Dirichlet character.

Cf. for fundamental discriminants -3..-24: A227128, A395768, A395771, A395769, A395772, A395773, A395774, this sequence, A395776, A395777.

Sequence in context: A199088 A293974 A372672 * A346036 A138557 A129303

Adjacent sequences: A395772 A395773 A395774 * A395776 A395777 A395778

Keyword

nonn,easy,mult

Author

Jianing Song, May 05 2026

Status

approved